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Unformatted text preview: University of Texas at Dallas Midterm Examination #2 Last Name: First Name and Initial: ...................................... ............................................ Course Name: Mathematical Analysis 1 Number: MATH 4301 Section: 001 Instructor: Wieslaw Krawcewicz Date: November 10, 2011 Duration: 1:15 hours E-mail Address: ....................................... Student’s Signature: . .......................................... . 1 Metric Spaces and Topological Concepts 1. Definition of metric 2. Definition of metric space 3. Open balls in metric space and open sets 4. Definition of a norm ∥ · ∥ on a vector space V and the associated with this norm metric. 5. Properties of open sets in a metric space ( X,d ). 6. Definition of a closed set in a metric space ( X,d ). Properties of closed sets. 7. Definitions of interior, boundary and closure of a set A (using logical sequences and quantifiers). Properties of these operations on sets. 8. Definition of a convergent sequence in a metric space (using quantifiers). 9. Prove that a sequence in a metric space can have only one limit. 10. Definition of a Cauchy sequence in a metric space. Show that every convergent sequence in a metric space is Cauchy and give an example of Cauchy sequence that doesn’t converge. 11. When a function f : X → X is called Lipschitzian? When a Lipschitzian function is called contraction ? 12. When a metric space ( X,d ) is called complete? Give an example of a complete metric space. Give an example of a metric space that is not complete. 13. State the Banach Contraction Principle. 14. When a point x o ∈ X is called a limit point of a set A in a metric space ( X,d )? When a point x o ∈ A is called an isolated point of A . Prove that x o is a limit point of A if and only if ∃ { x n }⊂ A x n ̸ = p ∧ lim n →∞ x n = x o . 15. What is the set A ′ (the set of all limit points of A – give a formal definition of A ′ ). 16. Give an example of a sequence { x n } in R containing three subsequences convergent to different limits. Can you construct such a sequence in a way that x k ̸ = x n for k ̸ = n . 17. Use the definition of a convergent sequence and Cauchy sequence in a metric space ( X,d ) to show that if a Cauchy sequence contains a convergent subsequence, then it is conver- gent. 18. Prove that every bounded sequence in R contains a convergent subsequence. 19. What are the compact sets in R n . Give examples of compact sets in R 2 and also of the sets which are not compact. 20. Let ( X,d ), ( Y,d ′ ) be two metric spaces and f : X → Y a function. Use the notion of the limit of a function to define when f is continuous at a point a ∈ X . When the function f is called continuous?...
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